Final answer:
To find the probability that the sample mean is between 85 and 92, we can use the z-score formula. The probability that the sample mean is between 85 and 92 is approximately 0.7015, or 70.15%.
Step-by-step explanation:
To find the probability that the sample mean is between 85 and 92, we can use the z-score formula. The z-score measures the number of standard deviations a value is away from the mean in a normal distribution. In this case, the sample mean is our value of interest. The formula for the z-score is z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Using the given information, we have μ = 90, σ = 15, and n = 25. Plugging these values into the formula, we get z = (85 - 90) / (15 / √25) ≈ -1.67 and z = (92 - 90) / (15 / √25) ≈ 0.67. Now we need to find the area under the standard normal curve between these two z-scores.
Using a standard normal distribution table or a calculator, we find that the area to the left of -1.67 is approximately 0.0475 and the area to the left of 0.67 is approximately 0.749. Therefore, the probability that the sample mean is between 85 and 92 is approximately 0.749 - 0.0475 = 0.7015, or 70.15%.